I have a function with a constant term upfront and was wondering how to obtain the complex conjugate. The function is say:
$$(g(z))^* = (\frac{-i\Gamma}{2\pi}\times f(z))^* $$
Would the conjugation be conducted 'distributively' such that:
$$ \ \ \ \ \ \ \ \ \ \ \ \ (g(z))^* = (\frac{-i\Gamma}{2\pi})^*\times f(z)^* $$ $$ \rightarrow(g(z))^* = \frac{i\Gamma}{2\pi}\times f(z)^* $$
also in general for a complex function if a complex function is contained within another how is the conjugate computed, like this?
$$g(z) = h(f(z)) $$ $$ \rightarrow g(z)^* = (h(f(z)))^* $$ $$ \rightarrow g(z)^* = h^*(f^*(z)) $$
Sorry for the dual question , but thank you all for your time!
Concerning your first question, you are right. This follow from the equality $\overline{z\times w}=\overline z\times\overline w$.
Now, if $f$ is a function from $\mathbb C$ into itself, let us define $\overline f(z)$ as $\overline{f(z)}$. Then $\overline{h\circ f}=\overline h\circ f$, which, in general, is not the same thing as $\overline h\circ\overline f$.